3.9.45 \(\int \frac {e^{-e^{\frac {1}{3} (5 x+x^2)}} (6 x^2+e^{e^{\frac {1}{3} (5 x+x^2)}} (3+3 x+3 e^x x)+e^{\frac {1}{3} (5 x+x^2)} (-5 x^3-2 x^4))}{3 x} \, dx\)

Optimal. Leaf size=26 \[ -5+e^x+x+e^{-e^{\frac {1}{3} x (5+x)}} x^2+\log (x) \]

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Rubi [F]  time = 4.10, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {e^{-e^{\frac {1}{3} \left (5 x+x^2\right )}} \left (6 x^2+e^{e^{\frac {1}{3} \left (5 x+x^2\right )}} \left (3+3 x+3 e^x x\right )+e^{\frac {1}{3} \left (5 x+x^2\right )} \left (-5 x^3-2 x^4\right )\right )}{3 x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(6*x^2 + E^E^((5*x + x^2)/3)*(3 + 3*x + 3*E^x*x) + E^((5*x + x^2)/3)*(-5*x^3 - 2*x^4))/(3*E^E^((5*x + x^2)
/3)*x),x]

[Out]

E^x + x + Log[x] + 2*Defer[Int][x/E^E^((x*(5 + x))/3), x] - (5*Defer[Int][E^((-3*E^((x*(5 + x))/3) + 5*x + x^2
)/3)*x^2, x])/3 - (2*Defer[Int][E^((-3*E^((x*(5 + x))/3) + 5*x + x^2)/3)*x^3, x])/3

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \frac {e^{-e^{\frac {1}{3} \left (5 x+x^2\right )}} \left (6 x^2+e^{e^{\frac {1}{3} \left (5 x+x^2\right )}} \left (3+3 x+3 e^x x\right )+e^{\frac {1}{3} \left (5 x+x^2\right )} \left (-5 x^3-2 x^4\right )\right )}{x} \, dx\\ &=\frac {1}{3} \int \frac {e^{-e^{\frac {1}{3} x (5+x)}} \left (6 x^2+e^{e^{\frac {1}{3} \left (5 x+x^2\right )}} \left (3+3 x+3 e^x x\right )+e^{\frac {1}{3} \left (5 x+x^2\right )} \left (-5 x^3-2 x^4\right )\right )}{x} \, dx\\ &=\frac {1}{3} \int \left (-e^{-e^{\frac {1}{3} x (5+x)}+\frac {1}{3} x (5+x)} x^2 (5+2 x)+\frac {3 e^{-e^{\frac {1}{3} x (5+x)}} \left (e^{e^{\frac {1}{3} x (5+x)}}+e^{e^{\frac {1}{3} x (5+x)}} x+e^{e^{\frac {1}{3} x (5+x)}+x} x+2 x^2\right )}{x}\right ) \, dx\\ &=-\left (\frac {1}{3} \int e^{-e^{\frac {1}{3} x (5+x)}+\frac {1}{3} x (5+x)} x^2 (5+2 x) \, dx\right )+\int \frac {e^{-e^{\frac {1}{3} x (5+x)}} \left (e^{e^{\frac {1}{3} x (5+x)}}+e^{e^{\frac {1}{3} x (5+x)}} x+e^{e^{\frac {1}{3} x (5+x)}+x} x+2 x^2\right )}{x} \, dx\\ &=-\left (\frac {1}{3} \int e^{\frac {1}{3} \left (-3 e^{\frac {1}{3} x (5+x)}+5 x+x^2\right )} x^2 (5+2 x) \, dx\right )+\int \left (1+e^x+\frac {1}{x}+2 e^{-e^{\frac {1}{3} x (5+x)}} x\right ) \, dx\\ &=x+\log (x)-\frac {1}{3} \int \left (5 e^{\frac {1}{3} \left (-3 e^{\frac {1}{3} x (5+x)}+5 x+x^2\right )} x^2+2 e^{\frac {1}{3} \left (-3 e^{\frac {1}{3} x (5+x)}+5 x+x^2\right )} x^3\right ) \, dx+2 \int e^{-e^{\frac {1}{3} x (5+x)}} x \, dx+\int e^x \, dx\\ &=e^x+x+\log (x)-\frac {2}{3} \int e^{\frac {1}{3} \left (-3 e^{\frac {1}{3} x (5+x)}+5 x+x^2\right )} x^3 \, dx-\frac {5}{3} \int e^{\frac {1}{3} \left (-3 e^{\frac {1}{3} x (5+x)}+5 x+x^2\right )} x^2 \, dx+2 \int e^{-e^{\frac {1}{3} x (5+x)}} x \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 25, normalized size = 0.96 \begin {gather*} e^x+x+e^{-e^{\frac {1}{3} x (5+x)}} x^2+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6*x^2 + E^E^((5*x + x^2)/3)*(3 + 3*x + 3*E^x*x) + E^((5*x + x^2)/3)*(-5*x^3 - 2*x^4))/(3*E^E^((5*x
+ x^2)/3)*x),x]

[Out]

E^x + x + x^2/E^E^((x*(5 + x))/3) + Log[x]

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fricas [A]  time = 0.85, size = 36, normalized size = 1.38 \begin {gather*} {\left (x^{2} + {\left (x + e^{x} + \log \relax (x)\right )} e^{\left (e^{\left (\frac {1}{3} \, x^{2} + \frac {5}{3} \, x\right )}\right )}\right )} e^{\left (-e^{\left (\frac {1}{3} \, x^{2} + \frac {5}{3} \, x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((3*exp(x)*x+3*x+3)*exp(exp(1/3*x^2+5/3*x))+(-2*x^4-5*x^3)*exp(1/3*x^2+5/3*x)+6*x^2)/x/exp(exp(1
/3*x^2+5/3*x)),x, algorithm="fricas")

[Out]

(x^2 + (x + e^x + log(x))*e^(e^(1/3*x^2 + 5/3*x)))*e^(-e^(1/3*x^2 + 5/3*x))

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giac [B]  time = 0.70, size = 73, normalized size = 2.81 \begin {gather*} {\left (x^{2} e^{\left (\frac {1}{3} \, x^{2} + \frac {5}{3} \, x - e^{\left (\frac {1}{3} \, x^{2} + \frac {5}{3} \, x\right )}\right )} + x e^{\left (\frac {1}{3} \, x^{2} + \frac {5}{3} \, x\right )} + e^{\left (\frac {1}{3} \, x^{2} + \frac {5}{3} \, x\right )} \log \relax (x) + e^{\left (\frac {1}{3} \, x^{2} + \frac {8}{3} \, x\right )}\right )} e^{\left (-\frac {1}{3} \, x^{2} - \frac {5}{3} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((3*exp(x)*x+3*x+3)*exp(exp(1/3*x^2+5/3*x))+(-2*x^4-5*x^3)*exp(1/3*x^2+5/3*x)+6*x^2)/x/exp(exp(1
/3*x^2+5/3*x)),x, algorithm="giac")

[Out]

(x^2*e^(1/3*x^2 + 5/3*x - e^(1/3*x^2 + 5/3*x)) + x*e^(1/3*x^2 + 5/3*x) + e^(1/3*x^2 + 5/3*x)*log(x) + e^(1/3*x
^2 + 8/3*x))*e^(-1/3*x^2 - 5/3*x)

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maple [A]  time = 0.06, size = 21, normalized size = 0.81

method	result	size
risch	\(x +\ln \relax (x )+{\mathrm e}^{x}+x^{2} {\mathrm e}^{-{\mathrm e}^{\frac {\left (5+x \right ) x}{3}}}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*((3*exp(x)*x+3*x+3)*exp(exp(1/3*x^2+5/3*x))+(-2*x^4-5*x^3)*exp(1/3*x^2+5/3*x)+6*x^2)/x/exp(exp(1/3*x^2
+5/3*x)),x,method=_RETURNVERBOSE)

[Out]

x+ln(x)+exp(x)+x^2*exp(-exp(1/3*(5+x)*x))

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maxima [A]  time = 0.65, size = 23, normalized size = 0.88 \begin {gather*} x^{2} e^{\left (-e^{\left (\frac {1}{3} \, x^{2} + \frac {5}{3} \, x\right )}\right )} + x + e^{x} + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((3*exp(x)*x+3*x+3)*exp(exp(1/3*x^2+5/3*x))+(-2*x^4-5*x^3)*exp(1/3*x^2+5/3*x)+6*x^2)/x/exp(exp(1
/3*x^2+5/3*x)),x, algorithm="maxima")

[Out]

x^2*e^(-e^(1/3*x^2 + 5/3*x)) + x + e^x + log(x)

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mupad [B]  time = 0.17, size = 23, normalized size = 0.88 \begin {gather*} x+{\mathrm {e}}^x+\ln \relax (x)+x^2\,{\mathrm {e}}^{-{\left ({\mathrm {e}}^{x^2}\right )}^{1/3}\,{\left ({\mathrm {e}}^x\right )}^{5/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-exp((5*x)/3 + x^2/3))*((exp(exp((5*x)/3 + x^2/3))*(3*x + 3*x*exp(x) + 3))/3 + 2*x^2 - (exp((5*x)/3 +
x^2/3)*(5*x^3 + 2*x^4))/3))/x,x)

[Out]

x + exp(x) + log(x) + x^2*exp(-exp(x^2)^(1/3)*exp(x)^(5/3))

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sympy [A]  time = 76.10, size = 24, normalized size = 0.92 \begin {gather*} x^{2} e^{- e^{\frac {x^{2}}{3} + \frac {5 x}{3}}} + x + e^{x} + \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((3*exp(x)*x+3*x+3)*exp(exp(1/3*x**2+5/3*x))+(-2*x**4-5*x**3)*exp(1/3*x**2+5/3*x)+6*x**2)/x/exp(
exp(1/3*x**2+5/3*x)),x)

[Out]

x**2*exp(-exp(x**2/3 + 5*x/3)) + x + exp(x) + log(x)

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